You should not see the bag until ZIH. Customs for checked bags happens "behind the scenes" while you will put carry on luggage through an x-ray at customs just past baggage. Most likely your luggage will be checked in at LAX with a ZIH tag. Just confirm with them at that time.

If you are nervous about it (as I am sometimes), as you enter baggage claim, there is a small connections desk to your right, and they can let you know what terminal you are leaving from, and confirm that your bag will travel through,

I am currently seeing today's 10 am AM ZIH flight as leaving from T1. That desk should be able to give you instructions on getting the tram over to T1.